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You can take $\mathcal B(\varphi_1,\varphi_2)=\int \varphi_1(x)\ dx\ \int \varphi_2'(y) \ln|x-y|\ dy$ , i.e. $K$ is the derivative $\partial_y$ or $-\partial_x$ of (the distribution defined by) the locally integrable function $H(x,y)=\ln|x-y|$ .